Honestly, I don't believe the British would have abandoned Iraq, Transjordania or Egypt to the Axis so easily. They had suffered defeats there, but were NOT beaten, and still controlled most of the territory (the Axis was still far from Suez and hadn't taken Cairo, and wasn't in control of Transjordania or the south of Iraq either).
The British would also have eventually overwhelmed the Germans and Italians through sheer numbers (of British, Indian and other colonial soldiers, as well as RAF fighters and bombers) in both Africa and Middle East, if they had kept fighting. Logistics were on their side.
If the fighting had continued for a year or two, the Axis would have been driven out of Iraq, Jordania, and then Syria and Lebanon (as they could hardly reinforce those areas much, given the distances), and Egyptian front would have stabilized, and then the following years, Libya would have fallen.
In the meantime, it would have turned into a giant resource sinkhole for Germany, Italy and Vichy France, hampering their efforts on the Eastern Front.
In short : the Axis was in position (diplomatically and military) to demand the recognition of their control of continental Europe (and Malta), as well as the return of Italian East Africa, and the British government had to concede that (under pressure from the US and its own people, as well as reality on the battleground)
but the British were in position to NOT give up Egypt, Levant or Iraq. And the USA would have backed Britain there.
And they have no reason to. Iraq is their oil supply, and if the Germans gain access to Iraq, they also come close to the Arabian peninsula and Iran (their other oil sources), which is a major problem as Iran was German-friendly. The Suez Canal is very important for British civilian shipping, and a major income source.
And abandoning Europe is a major humiliation (that can still be stomached now), but abandoning Middle East too is a double humiliation.